Recursive kinematics and dynamics for parallel structured closed-loop multibody systems

A kinematic formulation for the parallel structured closed-loop multibody mechanical systems, such as Stewart Platform and Hexapod machine tools, is presented in this paper, which is recursive in nature. This also leads to the minimum order representation of the dynamic equations of motion. The recursive algorithms are known for their efficiency when a system is large. They also provide many physical interpretations. On the other hand, the minimum order dynamic equations of motion are desired in control and simulation. For the latter a minimum set of dynamic equations of motion leads to a numerically stable integration algorithm that does not violate the kinematic constraints. Two recursive algorithms, one for the inverse and another for the forward dynamics, are proposed. The overall complexity of either problem is O(n) + nO(m), where n and in are the number of legs and the total number of rigid bodies in each leg, respectively. Hence the proposed formulation exploits the advantages of both minimum order representation and recursive algorithms, which earlier were available only for the open-loop systems such as serial manipulators. The method is illustrated with three examples: a one-degree-of-freedom (DOF) slider-crank mechanism, four-bar linkage, and a two-DOF five-bar planar parallel manipulator.